1. Introduction

Gain which provides the mean to overcome losses is always the focuses of research. Light amplification could be realized under the condition that losses of photon have been sufficiently compensated by gain, giving rise to the generation of laser [1]. Finding an effective method to implement control of gains is of significant importance both for fundamental researches and practical applications. Modern communication systems introduce the optical gains by adopting the erbium-doped fiber amplifier technology to achieve high-performance information transmission [2]. Ultra-sensitive sensing can be well realized by adding proper gain to compensate the intrinsic loss [3]. Through introducing balanced gain and loss, parity-time-symmetric systems [4] pave the new way towards non-reciprocal propagation devices [5–7], ultra-low threshold phonon laser [8] and lasing control [9–11].

Whispering-gallery-mode (WGM) microresonators [12–14], which can greatly enhance the light-matter interaction due to its ultra-high quality factor Q and very small mode volume, have got lots of attention in the last decade [15]. More interestingly, their intrinsic loss can be well compensated through introducing optical gain provided by the rare earth ions [16–18] or the Raman gain [19]. The WGM microresonators with optical gains are also called active WGM microresonators. They have great performances in a wide range of researches and applications, such as the control of the coupling properties [20], optical amplifiers [17], ultra-sensitive particle sensing [3, 21], ultralong photon storage [22] and all-optical diode [5]. Therefore, the control of gains in active WGM microresonators becomes very important, and it has been paid many attentions in recent years.

Up to date, the control of gains in active WGM microresonators are realized through changing the power of the pump. For example, in the pump-probe experiment of Er³⁺-doped WGM microcavities, the gain for the probe signal around 1550 nm band can be well modified by changing the pump around 1440 nm band [5, 17]. Here, we proposed that by introducing an auxiliary control signal which is also around 1550 nm band in Er³⁺-doped WGM microcavities, it is possible to realize the control of gains for the probe signal. Our experiment demonstrated that optical gain for the probe signal can be well modified by this control signal while the pump is kept unchanged. When we tune the overlap between the control and probe signals in time domain, the transition of Lorentz peak, Fano-like resonance and Lorentz dip for the probe signal can be realized from the transmission spectra.

To better explain the underlying physics behind these phenomena, we build a model based on the coupled-mode theory and laser rate equations, which can fit well with the experimental results. Optical gains during this resonance control process are further analyzed in detail. We found that gain competition between the probe and the control signal plays an important role during this resonance control.

2. Experimental setup and methods

Experimental setup and basic principles to control resonance lineshapes (or optical gains) in multimode Er³⁺-doped WGM microcavities are demonstrated in Fig. 1. As shown in Fig. 1(a), a tunable laser with emission in 1440 nm band works as the pump. Signal lights are provided by two external-cavity tunable lasers with linewidth less than 200 KHz in 1534 nm (probe signal s₁) and 1552 nm (control signal s₂), which coincide with the emission spectra of erbium ions. The pump, which is thermally locked, is tuned on resonance with microtoroid [23], but under the lasing threshold, to provide sufficient optical gain. Two triangle waves generated by the function generator are utilized to implant fine scan of the wavelengths around the center resonances. Frequency-scanning speeds of the signals can be adjusted by changing the amplitudes and frequencies of these two triangle waveforms. Variable optical attenuators are used to control the input power. A single mode optical taper fiber with waist diameter about 1–2 μm couples light in and out of the active microtoroid [24]. The coupling strength between the microtoroid and taper fiber can be efficiently tuned by precisely controlling their gap through a three-axis stage with 100 nm resolution. Light coupled out from the taper fiber is divided into two ports: 10% port is connected with optical spectrum analyzer (OSA), while 90% port is divided into 1440 nm, 1534 nm and 1552 nm band through wavelength-division-multiplexer (WDM). Transmitted signals through WDM are received by photodetectors (PDs with bandwidth less than 125 MHz) and reflected on the oscilloscope.

 

Fig. 1 (a) Schematics of the experimental setup. VOA: variable optical attenuator; WDM: wavelength division multiplexer; PD: photodetector; OSA: optical spectrum analyzer; and PC: polarization controller; (b) Illustration describing the way to tune overlap between the probe and control signals in time domain; (c) Energy levels of erbium.

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The sample we use is Er³⁺-doped microtoroid, which is fabricated upon silicon wafer containing 2 μm Er³⁺ dopants sol-gel silica. After photolithography, pattern transfer, dry etching and reflow, surface-tension-induced toroid-shape forms [13]. Optical gain for the corresponding signal can be characterized through the transmission spectra. For example, when signal s₂ is turned off, the transmission spectra of signal s₁ can exhibit Lorentz dip, Fano-like resonance and Lorentz peak in Fig. 2(c) with the increasing pump power. This is attributed to the time-dependent optical gain and the gain increases with the pump, which was reported and well explained in the previous work [25]. As reported in Ref. [25], this time-dependent gain also depends on the scan speed and the input power of the probe signal. When the scan speed of the probe is increased or the input power of the probe is decreased, the transition of Lorentz dip, Fano-like resonance and Lorentz peak can also be observed. In fact, this Fano-like resonance is different from the traditional Fano resonance which results from the interference, such as the interference between a discrete state and the continuum [26, 27], the interference between optical modes or the mechanical modes [28–31]. The Fano-like resonance here originates from the time-varying gain instead of the interference between different optical paths. It should be noted that we needed to find the suitable coupling position to couple both the pump and the probe into the microresonator in Fig. 2(c), the coupling condition between the microresonator and the taper fiber is different from that when we measuring the lasing threshold in Fig. 2(b). Besides, for the situation when two signals co-exist, we can tune their overlap in time domain by adjusting the relative phase between two triangle waves, as shown in Fig. 1(b).

 

Fig. 2 (a) Emission spectrum of Er³⁺-doped microtoroid with pump in 1440 nm; (b) Lasing characteristics for optical modes ${\lambda }_s{}_{{}_1}=1534\text{nm}$ and ${\lambda }_s{}_{{}_2}=1552\text{nm}$; (c) Responses in transmission spectra of signal s₁ with increasing pump power 0 μW, 31 μW and 74 μW when signal s₂ is turned off.

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3. Experimental results

Three optical modes of microtoroid were characterized firstly. With Er³⁺ concentration around 7.5 × 10¹⁸ ions/cm³ and major diameter 35 μm, the microtoroid has intrinsic quality factor Q ~ 3.2×10⁷ in the 1440 nm pump mode and 5.4×10⁶, 9.2 × 10⁵ in 1534 nm, 1552 nm mode respectively. Figure 2(a) shows the emission spectrum of Er³⁺-doped microtoroid. Figure 2(b) demonstrates corresponding lasing power as a function of the pump. The corresponding lasing thresholds are 54 μW for the mode ${\lambda }_s{}_{{}_1}=1534\text{nm}$ and 63 μW for the mode ${\lambda }_s{}_{{}_2}=1552\text{nm}$ under the best coupling conditions.

Experimental results of the resonances control based on the gain competition are demonstrated in Fig. 3. Signal s₂ with center wavelength in 1552 nm works as the control signal, while signal s₁ with center wavelength in 1534 nm is the probe signal. The input power is 430 nW for the control signal s₂ and 291 nW for the probe signal s₁. Frequencies of the triangle waves implanted on the tunable lasers to generate these two signals are the same. We keep the probe signal s₁ fixed and tune the phase of the triangle wave implanted on the control signal s₂, like that in Fig. 1(b). Responses of these two signals are shown from Figs. 3(a) to 3(e). As seen in Fig. 3(a) and 3(e), when two resonances have no overlap in time domain entirely, probe signal s₁ exhibits a Lorentz peak and the control signal s₂ shows a Fano-like resonance due to its relatively lower quality factor Q and higher input power. From Fig. 3(b) to 3(d), probe signal undergoes Fano-like resonance, Lorentz dip and Fano-like resonance process, while the control signal has no obvious changes, suggesting that the optical gain for the probe signal s₁ can be well modulated by the control signal s₂.

 

Fig. 3 Experimental and simulation results of the normalized transmission spectra for probe signal s₁: red line with center wavelength at 1534 nm, input power 291 nW, and frequency-scanning speed 3.9 THz/s; and for control signal s₂: blue line with center wavelength at 1552 nm, input power 430 nW, and frequency-scanning speed 3.1 THz/s. (a)–(e) Responses of the transmission spectra when we tune the overlap between two signals in time domain. The right panels show zooms of the dashed line areas. Here, all the blue lines have been moved upward by 1.5 with respect to the normalized transmission. The parameters are R = 35 μm, A_p = 1.5 × 10⁻⁸ cm², $A_s{}_{{}_1}=1.5\times {10}^{-}{}^8{\text{cm}}^2$, $A_s{}_{{}_2}=1.55\times {10}^{-}{}^8{\text{cm}}^2$, n_s = 1.35, n_p = 1.35, ${\mathrm{\Gamma }}_{s_1}{\sigma }_{s_1}^a=4.1\times {10}^{-21}{\text{cm}}^2$, ${\mathrm{\Gamma }}_{s_1}{\sigma }_{s_1}^e=9.86\times {10}^{-21}{\text{cm}}^2$, ${\mathrm{\Gamma }}_{s_2}{\sigma }_{s_2}^a=5.5\times {10}^{-21}{\text{cm}}^2$, ${\mathrm{\Gamma }}_{s_2}{\sigma }_{s_2}^e=1.51\times {10}^{-21}{\text{cm}}^2$, ${\sigma }_p^a=7.9\times {10}^{-21}{\text{cm}}^2$, ${\sigma }_p^e=4.0\times {10}^{-21}{\text{cm}}^2$, N_total = 7.5 × 10¹⁸ ions/cm³, τ₂₁ = 6 ms and τ₃₂ = 38 μs.

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From the experimental results above, we demonstrated that the transmission spectra of the probe signal can be effectively controlled through introducing the auxiliary control signal. To better understand the physics behind this control process, we increased the input power of probe signal s₁ to 3.6 μW. Then we keep the control signal s₂ fixed and tune the phase of the triangle wave implanted on the probe signal s₁. The changes of the transmission spectra during this process are demonstrated in Fig. 4. As shown in Fig. 4(b), the peak of the control signal s₂ is largely suppressed by the probe signal s₁. Since the input power of s₁ is much higher than s₂, probe signal s₁ can not be effectively modulated by the control signal s₂ any more. On the contrary, the control signal s₂ is largely suppressed by the probe signal s₁. It is worth noting that the frequency-scanning speeds of these two singles were increased by two times in Fig. 4 relative to that in Fig. 3 so as to make the results more apparent.

 

Fig. 4 (a)–(d) Experimental and simulation results of the transmission spectra when tuning the overlap of two signals in time domain. Right panels are zooms of the dashed line areas. Signal s₁: red line, input power 3.5 μW, and frequency-scanning speed 7.8 THz/s. Signal s₂: blue line, input power 430 nW, and frequency-scanning speed 6.2 THz/s. Here, all the blue lines have been moved upward by 1 with respect to the normalized transmission. The other parameters are the same with those in Fig. 3.

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4. Theoretical model and simulation

As we know, the interference between different optical modes [26–30] could lead to the appearance of Fano resonance or other lineshapes in the transmission spectra. However, by successfully control the polarizations of two signals as well as the coupling position of the taper fiber [32] in our experiment, the changes in transmission spectra can not be simply explained by the interference between different optical modes. Furthermore, since the control and probe signals share the same population inversion provided by the pump, gain competition between them plays an important role during this resonance control process.

Based on the coupled-mode theory, evolution of the intracavity optical fields are governed by the following differential equations [33]

Here, a_p and $a_s{}_{{}_1(s_2)}$ represent the amplitudes of the pump and two signals. For simplification, counter-routing field induced by Rayleigh scattering is not included here. Δω_p and $\mathrm{\Delta }{\omega }_s{}_{{}_1(s_2)}$ account the detunings of the input lasers with respect to the cavity center resonances. The intrinsic energy decay rates are ${\kappa }_p^0$ and ${\kappa }_{s_1(s_2)}^0$ while ${\kappa }_p^{ext}$ and ${\kappa }_{s_1(s_2)}^{ext}$ describe the coupling rate into and outgoing cavities. g_p and $g_{s_1(s_2)}$ stand for the gains induced by the interaction between light and Er³⁺. The launched input power are denoted by ${\left|a_p^{in}\right|}^2$ and ${\left|a_{s_1(s_2)}^{in}\right|}^2$.

The variances in transmission spectra are determined by $g_{s_1(s_2)}$ because ${\kappa }_{s_1(s_2)}^0$ and ${\kappa }_{s_1(s_2)}^{ext}$ are nearly unchanged in our experiment. For previous pump-probe experiments, $g_{s_1(s_2)}$ is regarded as time-independent constants. Under this assumption, with the standard input-output relationship $a_{s_1(s_2)}^{out}=a_{s_1(s_2)}^{in}+\sqrt{{\kappa }_{s_1(s_2)}^{ext}}{a}_{s_1(s_2)}$ it is easily found that ${\kappa }_{s_1(s_2)}^0<g_{s_1(s_2)}$ gives rise to the Lorentz peak under the steady-state situation. In contrast, the Lorentz dip corresponds to the situation ${\kappa }_{s_1(s_2)}^0>g_{s_1(s_2)}$.

It is worth noting that the way regarding $g_{s_1(s_2)}$ as time-independent constants can not describe our experiments correctly because the frequencies of the input signals are always scanning. To better understand the dynamic behavior of $g_{s_1(s_2)}$, we adopt the three-energy level model which is often used in laser theory to describe the interaction between the active medium and light. In our model, the populations of Er³⁺ per unit volume in the ground state, metastable and intermediate state are represented by N₁, N₂ and N₃ respectively. As shown in Fig. 1(c), the transition of Er³⁺ from the ground state to the intermediate state can be realized through the excitation of the pump. Ions in the intermediate state will decay to the metastable state with spontaneous transition rate given by 1/τ₃₂. Meanwhile, optical gains for two signals are provided by the population inversion between the metastable state and the ground state. The effective intracavity gain coefficients can be expressed as [2,34]

Here, C_i = λ_i/(2πRA_in_ih),i = s₁,s₂, p. R and A_i represent the effective radius and cross area of the corresponding optical mode. h is the Planck constant. 1/τ₂₁ describes the spontaneous emission decay rate from the metastable state to the ground state. The build-up time of gain provided by the decay of Er³⁺ from the intermediate state to the metastable state is τ₃₂. It means that the gain consumed by signals needs time τ₃₂ to recover, which can be reflected from the transmission spectra.

Next, we solve Eqs. (1)–(6) numerically. The simulation results which can fit well with the experimental data are demonstrated in Fig. 3 and Fig. 4. Our simulations show that the three-energy level model and the way we take gains as time-dependent can well describe this dynamic processes. Meanwhile, time-dependent gains during the resonance control process are shown in Fig. 5 and Fig. 6. The red and blue solid lines represent the optical gains for signal s₁ and s₂, while the dashed lines indicate the intrinsic energy decay rates ${\kappa }_{s_1(s_2)}^0$. Since the input signals are scanned around the resonance frequencies, we show the changes of these dynamic gains in time domain.

 

Fig. 5 Simulation results for the corresponding optical gains of the probe signal s₁ (red lines) and the control signal s₂ (blue lines) using the parameters in Fig. 3. Black and green dashed lines indicate the intrinsic decay rates. (a)–(e) and (f)–(j) correspond to the gains of two signals in Figs. 3(a)–3(e) respectively.

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Fig. 6 Simulation results for the corresponding optical gains of the probe signal s₁ (red lines) and the control signal s₂ (blue lines) using the parameters in Fig. 4. The dashed lines indicate the intrinsic decay rates. (a)–(d) and (e)–(h) correspond to the gains of two signals in Figs. 4(a)–4(e) respectively.

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As shown in Fig. 3(a), when these two signals have no overlap in time domain, the control signal s₂ is the Fano-like resonance while the probe signal s₁ is a Lorentz peak. It can be well explained in Figs. 5(a) and 5(f). With signal s₂ coupled into microcavities, gain provided by the transition of Er³⁺ from the metastable state to the ground state will be consumed quickly. Subsequently, time-varying gain for s₂ is not sufficient to compensate the loss while the scanning s₂ is still around resonance. This is due to the slower build-up rate (1/τ₃₂) of the gain relative to the frequency-scanning speed of signal s₂. On the whole, the transmission spectrum of the control signal s₂ exhibits Fano-like asymmetric resonance. However, the situation of signal s₁ is different because the optical mode which s₁ is scanning around has higher quality factor (or narrower linewidth). Thus, signal s₁ will pass this optical mode quickly and do not have enough time to undergo the gain-decrease process. Besides, signal s₁ has lower input power, namely less population inversion consumption. All of these give rise to the standard Lorentz peak for s₁. Next, after we tune the relative position between two signals in Fig. 3(b), probe signal s₁ becomes a Fano-like resonance. Here, population inversion between the metastable state and the ground state is mainly consumed by control signal s₂ due to its higher input power and lower quality factor of the optical mode which s₂ is scanning around. This results in little gain provided for s₁ in Fig. 5(g) and also the Fano-like resonance. Then, probe signal s₁ evolves into a standard Lorentz dip, as shown in Fig. 3(c). Since the population inversion consumed by signal s₂ previously has not recover entirely, corresponding gain for s₁ has an obvious decrease which is certified in Fig. 5(h). Thus, the Lorentz dip of probe signal s₁ appears. When we further tune the overlap between control signal and probe signal in time domain, probe signal s₁ can get more and more gain [Figs. 5(i) and 5(j)], resulting in the Fano-like resonance in Fig. 3(d) and the Lorentz peak in Fig. 3(e).

With the input power of signal s₁ increased to 3.5 μW, transmission spectrum of the control signal s₂ is largely suppressed when these two signals overlap in Fig. 4(b). Since population inversion of Er³⁺ is mainly consumed by signal s₁ due to its higher input power, control signal s₂ can not implant effective control of gain on probe signal s₁ any more. Thus, optical gain for control signal s₂ has an dramatically decrease at the time when s₁ and s₂ overlaps in Fig. 6(b) and this corresponds to the suppression of transmission in Fig. 4(b).

In our model, τ₃₂ plays an important role during the resonance control process. The time-dependent gains provided for the probe and control signals are determined by the populations of Er³⁺ ions in the metastable state, and the Er³⁺ ions in the metastable state are obtained through the decay from the intermediate state. The decay rate is 1/τ₃₂, and this can be proved in the recovery time of the gain for the probe signal in Figs. 5(f) and 5(j) as well as Figs. 6(e) and 6(h). If τ₃₂ is smaller, the probe and control signal can get more gains in time. For instance, the Fano-like resonance of the probe signal could evolve into the Lorentz peak if we decrease τ₃₂ according to our simulation. Besides, due to the existence of τ₃₂, the gains consumed by these two signals will not recover in time, making the gain competition between the probe and control signals become more obvious. Thus, this property can be used to achieve the resonance control.

From the analyses above, our theoretical model and simulations have given good explanations for experimental results. We also calculated time-varying gains of signals s₁ and s₂ in the experiment. Our analyses show that dynamic gain and the gain competition between control signal and probe signal play an important role during this resonance control process. Based on this, we can precisely control optical gains and the transmission spectra of probe signal by adding the auxiliary control signal while the pump is kept unchanged. This method is totally different from previous methods which achieve this goal by changing the pump. The method through changing the pump is simple and effective. However, since the input power of the pump is much larger than the probe, the changes of the pump can significantly affect the properties of microresonator, like the temperature. It may result in the shift of resonant wavelength and introduce other unwanted effects. Instead, the auxiliary signal has relatively lower input power. When this control signal is tuned, the unwanted effects generated from this tuning can be significantly reduced. Thus, this method can be used in precisely controlling the optical gain without introducing additional noise or unwanted effects into the system. For instance, we can use this method in parity-time-symmetric system consisting of the coupled microresonators with gain and loss. The transition between the parity-time broken regime and unbroken regime can be well controlled through modifying the gain in the active microresonator using this method. The active microresonator will not have obvious resonance shift using this method relative to the traditional pump control so that the coupling condition between these two microresonators can be kept unchanged.

5. Conclusion

In summary, we have put forward an effective and flexible way to achieve the modification of transmission spectra in Er³⁺-doped WGM microresonators without modifying the pump. Different from previous methods, auxiliary control signal is introduced which is also in the signal region (around 1550 nm band). The transmission spectra of the probe signal, namely the optical gain, can be effectively controlled without changing the pump. According to the tuning of the control signal, the Lorentz peak, Fano-like resonance and Lorentz dip lineshapes could be observed for the probe signal. The theoretical model based on the coupled-mode theory and laser rate equations is constructed which can well explain these results when the gains are taken as time-dependent. Further analysis points out that the dynamic gain as well as the gain competition play an important role in this control process. Our experiments not only provide a new and effective method to modify the transmission spectra, but also further prove the dynamic property of gains in Er³⁺-doped WGM microresonators, which may be useful in the exploration of the nonreciprocal optical devices.